Course Identification

Basic  group theory (up to the classification of finitely generated Abelian groups and notions of normal subgroups and conjugasy classes); 

Basic linear algebra (rank of a linear operator, classification of quadratic forms...) 

Basic analysis of functions of several variables (up to Implicit Function theorem and Taylor series),   

Basic topology of subsets of Euclidean spaces (notions of closed, open, compact, connected sets, continuous functions and their topological properties).

An introduction to basic facts and notions of algebraic topology:

topological spaces and operations on them, homeomorphism and homotopy equivalence, smooth manifolds, cell complexes, algebraic invariants of topological spaces (fundamental groups and other homotopy groups, homology and cohomology groups), exact sequences, coverings and fiber bundles, Morse theory, multiplication in cohomology. Familiarity with popular topological manifolds (surfaces, projective spaces and Grassmann manifolds...). The course can also serve as a preparation to the course of a more advanced algebraic topology to be read in the second semester.

One will be able  to calculate homotopy groups and cohomology groups and rings of manifolds, cell complexes and other interesting topological spaces; to prove or disprove topological equivalence of not too complicated topological spaces, to investigate smooth manifolds by methods of Morse theory.

 A.Hatcher, Algebraic Topology